Chapter 4: Projection of mortality

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By Patrice Dion, Nora Bohnert, Simon Coulombe and Laurent Martel

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Introduction

Mortality trends have been evolving slowly and in a generally linear fashion for almost a century. This consistent pattern facilitates projections of future death rates. Yet, in most countries, mortality projections have underestimated the rise in life expectancy (Lee and Miller 2001; Keilman 2007).

Expert opinion has also tended to provide fairly pessimistic views regarding future improvements in life expectancy (Booth and Tickle 2008; Lee and Miller 2001). These days, concerns about specific health issues such as obesity and diabetes lead some demographers to think that life expectancy at birth could stop its increasing trend (Olshansky et al. 2005). Still, others highlight the fact that past increases in life expectancy occurred in spite of some health issues such as widespread cigarette smoking (Shkolnikov et al. 2011).

Among the various components of population change used to formulate the population projections – that is, mortality, fertility and migration – mortality does not have the largest impact in terms of the total population. Unlike fertility or immigration, mortality generally has no compound effect on the future number of births. On the other hand, it does have a large impact on projections of the oldest ages of the population. While the main driver of the current population aging trend was fertility (Hyndman and Booth 2008), declining mortality trends at older ages in particular have intensified population aging and will continue to do so in the context of low and fairly stable fertility. Thus, plausible projections of mortality are of chief importance to informing welfare and public policy planning about future trends in population aging.

Mortality trends

Over the last century, the annual number of deaths in Canada has generally increased, reaching 242,100 in 2011 (Figure 4.1).Note 1 On the other hand, the crude death rate (the number of deaths per thousand persons) has fluctuated around 7.0 per thousand since the 1980s, following decreases between the 1950s and 1970s. The increase in the number of deaths over time can be attributed mostly to population growth but also to population aging. As the age structure of the population becomes older, a relatively larger proportion of the total population is found in the older age groups that experience higher rates of mortality.

Figure 4.1 Number of deaths and crude death rate, Canada, 1926 to 2011

Description for figure 4.1

Mortality trends by age and sex

Over the lifespan, mortality rates follow a pattern similar to a checkmark shape: the mortality rate is higher in the first year of life; mortality rates then decline to their lowest levels in childhood and slowly increase throughout adulthood, reaching their highest levels at the very oldest ages (Figure 4.2). As in the past, in 2011, females experienced lower mortality rates than males at all ages.

Figure 4.2 Logarithms of probabilities of dying by age and sex, Canada, 2009/2011

Description for figure 4.2

Life expectancy at birth is an indicator that is by nature strongly influenced by infant mortality trends. While reaching its lowest rate on record in 2011 (4.8 deaths per thousand live births), the infant mortality rate in Canada has been quite stable since the mid-1990s following a long period of decline. It is unlikely that this plateau of the infant mortality rate is a result of Canada approaching a ‘natural limit’, given that several other countries have posted lower rates in recent years.Note 2 Instead, it is likely that general reductions in infant mortality in Canada have been offset by various trends related to the increased prevalence of older mothers, in addition to increased recognition of birth registration requirements (Bohnert 2013).

The elevated risk of death for males relative to females in young adulthood began to emerge as a pattern in the 1950s and reached its highest levels in the late 1970s and early 1980s (Figure 4.3), mostly a result of the higher risk among young males of deaths due to accidents, violence and suicide (Milan and Martel 2008). Since the 1980s, there has been a reduction in the sex differential in the probability of death in young adulthood, primarily the result of a larger decrease in the number of deaths from accidents and violence for males than for females in recent years. Between 1981 and 2010, a decline also occurred in the sex ratio of the probability of death in mid-adulthood (ages 45 to 69).Note 3 This trend is in part because the behavior of women (and associated risks of death) has become more similar to that of men over the last 40 years, particularly in the case of smoking behavior (Bélanger et al. 2001).

Figure 4.3 Male to female ratio of the probability of dying by age, Canada, selected years

Description for figure 4.3

In Canada, life expectancy has been increasing steadily throughout the 20th century. Between 1981 and 2010, male life expectancy at birth increased 7.4 years, from 71.9 to 79.3 years. Female life expectancy at birth gained 4.6 years, rising from 79.0 to 83.6 years. The gap in life expectancy at birth between Canadian males and females narrowed to 4.3 years in 2010 from its peak of 7.4 years at the end of the 1970s (Figure 4.4).

Figure 4.4 Difference (in years) between female and male life expectancy at birth, Canada, 1945 to 2010

Description for figure 4.4

As seen in Figure 4.5, the average annual improvements in life expectancy for the period 1981 to 2010 have been higher for males than females at all ages. Particularly at ages 40 to 75, improvements for males have been on average more than 80% higher than for females of the same ages. For instance, male life expectancy at age 61 increased on average 0.9% per year during this period compared to an average increase of 0.5% for females of the same age. For males, the largest annual improvements have occurred during their early seventies (for example, a 1.1% annual increase at age 73) while improvements for females were largest during their late seventies (a 0.6% annual increase at age 79, for example).

Figure 4.5 Average annual percentage change in life expectancy, by sex, Canada, 1981/1982 to 2009/2010

Description for figure 4.5

Mortality trends by region

As seen in Figure 4.6, there is no evidence that Canada’s provinces and territories are becoming more similar, in terms of life expectancy at birth, over time. The provinces, while much closer to one another in terms of life expectancy at birth than the territories, have actually experienced a slightly increasing divergence over time, particularly among males: in 1981, the highest male provincial life expectancy at birth was 2.8% higher than the lowest provincial life expectancy. In 2010, this differential had increased to 4.1%. When including the territories, there is no evidence of either convergence or divergence of life expectancy since 2000.

Figure 4.6 Difference (in percentage) between the highest and lowest life expectancy at birth, by sex, comparing the provinces only and the provinces and territories combined, 1981 to 2010

Description for figure 4.6

Among the provinces and territories, British Columbia experienced the highest life expectancy at birth for both females and males in 2010, as has been the case for several consecutive years. The variation in life expectancy at birth among the provinces and territories was larger among males (11.5 years) than among females (10.5 years) that year. For both sexes, the lowest life expectancies were found in Nunavut (73.9 years for females and 68.8 for males).

Since its advent, Nunavut has experienced considerably higher mortality rates compared to other provinces and territories, especially among young adults (see a comparison with the province of Ontario in Figure 4.7). There is some evidence that in recent years, among regions with a high proportion of Aboriginal residents, “premature mortality”Note 4 was twice as high in young adulthood (ages 15 to 24) compared to regions with a low proportion of Aboriginal residents, with injuries (mainly suicides and motor vehicle accidents) being the largest contributor to the relatively elevated number of deaths at younger ages (Allard et al. 2004).

Figure 4.7 Logarithm of probability of death by age and sex, Ontario and Nunavut, 2009/2011

Description for figure 4.7

Nunavut and the Northwest Territories also tend to have higher infant mortality rates than the Canadian average, resulting in a widening of the variation in these rates among Canada’s provinces and territories since the 1980s (Bohnert 2013).Note 5

International trends in mortality

The evolution of best-practice life expectancy—the highest life expectancy observed among national populations—is an indication of the path that non-leading countries could follow. Shkolnikov et al. (2011) found that the best-practice life expectancy for both the years 1870 to 2008 and the birth cohorts 1870 to 1920 have increased steadily over time.

The life expectancies of Canadian males and females have both been above the average of OECD countries in recent decades (Figure 4.8), though they have never reached the maximum.Note 6 Taking the respective OECD average life expectancy at birth of the two sexes, the gap has narrowed from 6.6 years in 1995 to 5.5 years in 2011. In comparison, the gap was smaller between Canadian males and females (4.3 years). In recent years, the life expectancy at birth of Canadian males has been closer to the leading country than that of Canadian females, reflecting the fact that Canadian male life expectancy improvements have been relatively strong over the last few decades. For example, in 2011, the maximum male life expectancy at birth, registered in Iceland, was 1.8% higher than that of Canadian males; while the maximum female life expectancy at birth, registered in Japan, was 2.8% higher than that for Canadian females.

Figure 4.8 Life expectancy at birth, Canada, OECD average and OECD maximum, by sex, 1995 to 2011

Description for figure 4.8

In many countries, a pattern referred to as ’rectangularization‘ or ’compression‘ of mortality has been observed to varying extents. Compression of mortality occurs when the proportion of persons in a population surviving to advanced ages increases. As a result, the survival curve increasingly takes on a rectangular shape as proportionally more mortality occurs at later and later ages. There is continuing debate as to whether complete rectangularization of mortality will eventually occur, meaning that all deaths would occur at roughly the same very advanced age. This would imply a fixed, predetermined biological limit to human survival (Manton and Singer 2002).

As seen in Figure 4.9, there is some evidence that mortality has become increasingly compressed (concentrated at older ages) in Canada. In 1931, 91.3% of males belonging to a synthetic cohort would have remained alive from birth to age one, compared to 99.5% of males in 2010. Similarly, in 1931, approximately three-quarters (75.2%) of males survived from birth to age 50, while by 2010, this proportion had increased to 97.3%. Similar improvements have occurred for females. While the curves in 2010 are more rectangular in shape than in 1931, the substantial extension in the length of the 2010 curves compared to those of 1931 suggests that Canada is not yet approaching a theoretical upper limit to life expectancy of a population.

Figure 4.9 Proportion (in percentage) of persons in a synthetic cohort surviving from birth to age x, by sex, Canada, 1931 and 2010

Description for figure 4.9

Alternative approaches to measuring average human longevity, such as the modal age of death, suggest that rectangularization or compression of mortality, while still present among males in most countries, is no longer occurring among females in many low mortality countries including Canada. Instead, evidence of a “shifting mortality regime”, whereby the majority of deaths shift to older and older ages, was found among females in Canada, the United States, France and Japan by Ouellette and Bourbeau (2011).

Ouellette and Bourbeau (2011) also find evidence that the long-lasting upward trend in the modal age of deaths slowed down substantially among Japanese men and has leveled off for Japanese women since the early 2000s. These recent developments could indicate that Japanese men and women—world leaders in human longevity—are approaching longevity limits in terms of modal lifespan. Indeed, Japanese female life expectancy at birth, while remaining the maximum registered among OECD countries, actually decreased slightly between 2009 (86.4 years) and 2011 (85.9 years). If, in the coming years, maximum female life expectancy were to stabilize, this might support theories that there is in fact an ultimate ‘ceiling’ or limit to human longevity. Many researchers, however, posit that future advancements in biomedical innovations and other genetic-environmental interactions could alter any genetically fixed limits to life expectancy, if indeed they might exist (Manton and Singer 2002).

Survey results

Results from the 2013 Opinion Survey on Future Demographic Trends suggest that Canadian demographers unanimously anticipate further improvement in life expectancy; however, differences emerged regarding the expected pace of this improvement and whether it might be of an indefinite or finite nature.

In the short term, respondents anticipate a considerable increase from the most recently observed levels of 78.6 years for males and 83.1 years for females, respectively: the median response of the most likely estimate of life expectancy at birth in 2018 is 80.6 years for males and 84.5 years for females (Figure 4.10). This would represent an increase of 2.5% for males and 1.7% for females from 2010. In comparison, the observed improvements between 2001 and 2010 were 3.5% and 2.0%, respectively.Note 7

Variation in the most probable estimates of life expectancy at birth in 2038 are relatively large, especially at the upper range of responses and more so for estimates of female life expectancy than for male life expectancy. This may reflects respondent knowledge of the fact that that previous projections (both in Canada and other countries) have consistently underestimated improvements in mortality (Lee and Miller 2001, Keilman 2007); consequently, respondents were perhaps more open to a wider range of possible improvements in the long-term future. For males, the estimates of the most likely life expectancy at birth range from 78.6 to 89.5 years while for females the range is between 83.1 and 94.5 years. Median values for the most probable estimates of life expectancy at birth in 2038 are 83.9 for males and 86.6 for females (Figure 4.10).

Figure 4.10 Summary statistics for the Opinion Survey on Future Demographic Trends regarding estimates of the life expectancy at birth, by sex, Canada, 2018 and 2038

Description for figure 4.10

Additionally, while respondents generally estimated a closing of the mortality gap between the sexes, there remained a sizeable gap estimated in 2038: taking the median responses of the most probable estimates, the gap between the sexes would be 3.9 years in 2018 and 3.3 years in 2038 compared to the most recently observed gap of 4.3 years in 2010.

To support their estimates, respondents most commonly mentioned the historical trend of continued decrease in mortality in Canada and other developed countries over the last century. In general, respondents mentioned that further advancements in medical technology, health care and health prevention could be expected; many also expected positive lifestyle changes at the population level related to smoking, diet and exercise. Respondents also mentioned factors that could slow the rate of mortality improvement in the future such as growing economic inequality and environmental changes.

Regarding mortality trends by sex, respondents generally expressed the view that there would be a continued convergence of male and female life expectancy in the future, but only to a certain point, which would be biologically fixed. In terms of age-specific mortality trends, little change in infant and child mortality was anticipated, while several respondents expected continued rectangularization of mortality and greater improvements at older ages.

Methodology

More than any other component of demographic growth, mortality lends itself to projections based on the extrapolation of past data. Indeed, life expectancy has been increasing steadily and generally following a linear trend, more so than trends for other demographic indicators such as the total fertility rate or the immigration rate. As in the past edition, the Li-Lee (2005) method (more details below in the “Method for coherent projections section) was used for the projection of future mortality rates of the different provinces and territories. Some improvements have been incorporated in this edition, the most notable being the implementation of the 'Extended Lee-Carter' model for modeling the evolution of the age patterns of mortality decline (Li et al. 2013).

Input data

Mortality data from 1981 to 2010 were used for the projection of future mortality rates. More specifically, age- and sex-specific death rates from Statistics Canada’s most recent life tables were used with some modifications.Note 8 The eventual life expectancies reached were not set in advance as they were determined by the extrapolation process (described below); however, the choice of the reference period has a large impact on the assumptions. The 1981 to 2010 period used in the projection of mortality was generally characterized by a decline in mortality and the steady narrowing of the gap in life expectancy at birth between males and females.

Note that using modeledNote 9 rates from the life tables rather than observed rates adds robustness to the trends while also addressing, to a certain extent, issues related to small numbers in some regions (of deaths, population or both) when projecting the individual provinces and territories separately. However, some further necessary adjustments were implemented. In the logic of Statistics Canada’s life table formulation, when missing values prevented the calculation of mortality rates at some ages, data from a higher-level geography was used (for example, if a mortality rate was missing at a certain age for Prince Edward Island, the rates for all of the Atlantic provinces combined was substituted).

Distinct procedures were used for the projection of mortality rates in the territories where the issues of small numbers and missing values were considerable. In order to obtain plausible base rates for the projection, special aggregate life tables were built for each territory made from the most recent 12 years of observed data. This procedure was, however, insufficient to model the rates after age 80. For this reason, the rates for ages 80 and over were set to follow what was observed at the Canada level at these ages during the same 12 years period, that is, the growth rate of mortality rates from one age to the next is “borrowed” from the age structure of mortality rates at the Canada level. A last step consisted of adjusting those rates so that the life expectancy at birth was identical to the life expectancy published in the abridged life tables for the territories for the latest year (2010). All these steps were designed to preserve as much as possible the very distinct patterns of mortality observed in each territory and that serve as the starting point for the projection. Note that the aggregation of data in a temporal perspective had no consequence on the modeling of past trends for the territories because this modeling was based strictly from data at the Canada level only, due to the same issues of small numbers and missing data (more details follow in the next section).

Method for coherent projections

Projection methods that place emphasis on limiting divergence between groups are typically labeled 'coherent'. Coherence is often considered preferable over divergence when it is expected that the factors which influence mortality trends are likely to affect all groups or regions in a country in a similar way, thus, limiting the extent of divergences. There are many reasons to believe that this reasoning should be applied to Canada. As seen earlier, there is no strong evidence of divergence (nor convergence) of life expectancy among the provinces and territories to date (Figure 4.6).

The Li-Lee method (Li and Lee 2005) was adapted from the commonly-used Lee-Carter method (Lee and Carter 1992) specifically to handle situations where coherence of the mortality projections of different groups is desired. The demonstrated robustness of the Lee-Carter method (Lee and Miller 2001; Booth 2006), its ability to project all provinces and territories in a coherent manner in its modified version by Li and Lee (2005), combined with its relative simplicity are key advantages for the projection of future mortality patterns in Canada. Specifically, the Lee-Carter method permits the projection of age and time patterns as two separate components. While the age component exhibits little variance over time, the time component is a highly linear time series that can be easily extrapolated. However, unlike the Lee-Carter method, the Li-Lee method limits the divergence of projections calculated for separate groups—in this case, the individual provinces and the territories—by using two components: a factor common to all provinces and territories and another factor specific to each.

As per the Li-Lee method, the log of age-specific mortality rates for each group (individual province or territory, sexes separated) is modeled as follows:

ln( m x,t )= a x,i + B x K t + b x,i k t,i + ε x,t,i (4.1) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeqabeabaa aabaGaciiBaiaac6gadaqadaqaaiaad2gadaWgaaWcbaGaamiEaiaa cYcacaWG0baabeaaaOGaayjkaiaawMcaaiabg2da9iaadggadaWgaa WcbaGaamiEaiaacYcacaWGPbaabeaakiabgUcaRiaadkeadaWgaaWc baGaamiEaaqabaGccaWGlbWaaSbaaSqaaiaadshaaeqaaOGaey4kaS IaamOyamaaBaaaleaacaWG4bGaaiilaiaadMgaaeqaaOGaam4Aamaa BaaaleaacaWG0bGaaiilaiaadMgaaeqaaOGaey4kaScccaGae8xTdu 2aaSbaaSqaaiaadIhacaGGSaGaamiDaiaacYcacaWGPbaabeaaaOqa aaqaaaqaaiaacIcacaaI0aGaaiOlaiaaigdacaGGPaaaaaaa@59BD@

where, a x,i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBa aaleaacaWG4bGaaiilaiaadMgaaeqaaaaa@39A3@  is the average of ln( m x,t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac6 gadaqadaqaaiaad2gadaWgaaWcbaGaamiEaiaacYcacaWG0baabeaa aOGaayjkaiaawMcaaaaa@3D31@ , at age x over T number of years (t=1,2,…,T), B x K t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOqamaaBa aaleaacaWG4baabeaakiaadUeadaWgaaWcbaGaamiDaaqabaaaaa@39E5@  represents the common factor model applied equally to all groups i, and b x,i k t,i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaBa aaleaacaWG4bGaaiilaiaadMgaaeqaaOGaam4AamaaBaaaleaacaWG 0bGaaiilaiaadMgaaeqaaaaa@3D61@  represents the model specific to each group i. Note that through projecting the logarithm of age-specific mortality rates, the possibility of obtaining negative rates is avoided.

In the common model, B MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOqaaaa@36BD@  is a vector of coefficients quantifying the change in the death rate at all ages associated with the scalar K t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBa aaleaacaWG0baabeaaaaa@37EB@ , the general time trend parameter. For a given sex, the common model applies to Canada. The first step consists of applying singular value decomposition (SVD) to a matrix A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaaaa@36BC@  whose elements a x,t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBa aaleaacaWG4bGaaiilaiaadshaaeqaaaaa@39AE@ are calculated as ln( m x,t )= a x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac6 gadaqadaqaaiaad2gadaWgaaWcbaGaamiEaiaacYcacaWG0baabeaa aOGaayjkaiaawMcaaiabg2da9iaadggadaWgaaWcbaGaamiEaaqaba aaaa@4046@ . The SVD is used as a technique of data reduction to obtain, from the matrix A, the first-order vectors K MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4saaaa@36C6@  and B MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOqaaaa@36BD@ , with the constraint that the sum of all elements of B MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOqaaaa@36BD@  must equal one, and the sum of all elements of K MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4saaaa@36C6@  must equal 0.Note 10 These constraints ensure that only one solution is derived from the SVD. The second step consists of adjusting the K t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBa aaleaacaWG0baabeaaaaa@37EB@  values from the K MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4saaaa@36C6@  vector to match the observed life expectancy. Then, in a third step, the K t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBa aaleaacaWG0baabeaaaaa@37EB@  values are extrapolated using the ARIMA time series method. Specifically, a Random Walk with Drift (RWD) process is used, which in this context is known for its good performance, simplicity and straightforward interpretation (Li et al. 2004):

K t = K t1 +d+ e t σ, e t N( 0 .1 ) (4.2) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeqabeabaa aabaGaam4samaaBaaaleaacaWG0baabeaakiabg2da9iaadUeadaWg aaWcbaGaamiDaiabgkHiTiaaigdaaeqaaOGaey4kaSIaamizaiabgU caRiaadwgadaWgaaWcbaGaamiDaaqabaGccqaHdpWCcaGGSaaccaGa e8hiaaIae8hiaaIae8hiaaIae8hiaaIae8hiaaIae8hiaaIae8hiaa Iae8hiaaIae8hiaaIae8hiaaIaamyzamaaBaaaleaacaWG0baabeaa kiab=XJi+jaad6eadaqadaqaaiaaicdacaGGSaGaaGymaaGaayjkai aawMcaaaqaaaqaaaqaaiaacIcacaaI0aGaaiOlaiaaikdacaGGPaaa aaaa@5728@

where d MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizaaaa@36DF@  is the drift term, a deterministic component reflecting the time trend, and σ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdmhaaa@37B9@  is a stochastic component, the standard deviation of random changes in K t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBa aaleaacaWG0baabeaaaaa@37EB@ . The projection of an age-specific mortality rate over n years at the Canada level, m x,t+n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaaBa aaleaacaWG4bGaaiilaiaadshacqGHRaWkcaWGUbaabeaaaaa@3B90@ , is calculated as:

e a x + B x K t (4.3) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeqabeabaa aabaGaamyzamaaCaaaleqabaGaamyyamaaBaaameaacaWG4baabeaa liabgUcaRiaadkeadaWgaaadbaGaamiEaaqabaWccaWGlbWaaSbaaW qaaiaadshaaeqaaaaaaOqaaaqaaaqaaiaacIcacaaI0aGaaiOlaiaa iodacaGGPaaaaaaa@419D@

The above model does not take into account specific mortality patterns in the provinces and territories. The calculation of these specific factors follows roughly the same logic as that of the common factor. In a first step, b x,i k t,i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaBa aaleaacaWG4bGaaiilaiaadMgaaeqaaOGaam4AamaaBaaaleaacaWG 0bGaaiilaiaadMgaaeqaaaaa@3D61@ , the specific factor for a given region i is computed by applying a SVD decomposition to the matrix of the residuals of the common factor model, each matrix entry being computed as:

ln( m x,t,i ) a x,i B x K t (4.4) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeqabeabaa aabaGaciiBaiaac6gadaqadaqaaiaad2gadaWgaaWcbaGaamiEaiaa cYcacaWG0bGaaiilaiaadMgaaeqaaaGccaGLOaGaayzkaaGaeyOeI0 IaamyyamaaBaaaleaacaWG4bGaaiilaiaadMgaaeqaaOGaeyOeI0Ia amOqamaaBaaaleaacaWG4baabeaakiaadUeadaWgaaWcbaGaamiDaa qabaaakeaaaeaaaeaacaGGOaGaaGinaiaac6cacaaI0aGaaiykaaaa aaa@4BF1@

The k t,i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AamaaBa aaleaacaWG0bGaaiilaiaadMgaaeqaaaaa@39A9@  values are then extrapolated using an ARIMA model, this time an auto-regressive model (AR1):

k t,i =c 0 i +c 1 i K t1,i + e t,i σ i , e t,i N( 0 .1 ) (4.5) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeqabeabaa aabaGaam4AamaaBaaaleaacaWG0bGaaiilaiaadMgaaeqaaOGaeyyp a0Jaam4yaiaaicdadaWgaaWcbaGaamyAaaqabaGccqGHRaWkcaWGJb GaaGymamaaBaaaleaacaWGPbaabeaakiaadUeadaWgaaWcbaGaamiD aiabgkHiTiaaigdacaGGSaGaamyAaaqabaGccqGHRaWkcaWGLbWaaS baaSqaaiaadshacaGGSaGaamyAaaqabaGccqaHdpWCdaWgaaWcbaGa amyAaaqabaGccaGGSaaccaGae8hiaaIae8hiaaIae8hiaaIae8hiaa Iae8hiaaIae8hiaaIae8hiaaIae8hiaaIae8hiaaIae8hiaaIaamyz amaaBaaaleaacaWG0bGaaiilaiaadMgaaeqaaOGae8hpI4NaamOtam aabmaabaGaaGimaiaacYcacaaIXaaacaGLOaGaayzkaaaabaaabaaa baGaaiikaiaaisdacaGGUaGaaGynaiaacMcaaaaaaa@638B@

where c 0 i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yaiaaic dadaWgaaWcbaGaamyAaaqabaaaaa@38B2@  and c 1 i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yaiaaig dadaWgaaWcbaGaamyAaaqabaaaaa@38B3@  are coefficients and σ i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaadMgaaeqaaaaa@38D3@  is the standard deviation of the model. The AR1 model allows the specific factor to eventually converge to a constant value (Li and Lee 2005).Note 11 Consequently, the specific factors create distinct patterns for each region that weaken over time, thus the achievement of coherent projections among the provinces and territories.

The Li-Lee method was used for the projection of mortality rates in all provinces. In the territories, it was deemed preferable to use only the common factor due to the small number of observations and small populations involved.Note 12

It should be noted that using this method, coherence is obtained between regions, but nothing is explicitly done to preserve some coherence between males and females. Although mortality rates are projected separately for males and females, results for the provinces, while not technically coherent, do not diverge as time goes by. Rather, they exhibit slow convergence in future life expectancy of males and females (as will be seen later), due to the fact that these trends were observed throughout the reference period chosen.Note 13 An adjustment was necessary in the territories however, without which the projected male life expectancy would eventually surpass that of females over the course of the projection. As this future trend is unlikely, the situation was remedied by setting the changes in mortality rates for males to adopt those projected for females in a gradual fashion over time.

Finally, the estimation of the B x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOqamaaBa aaleaacaWG4baabeaaaaa@37E6@  often lent negative values at old ages, which imply increasing mortality rates over time. While not impossible,Note 14 this is likely an artifact of the procedures used in the life tables to model mortality at old ages, necessitated by the presence of missing or volatile data. For this reason, the B x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOqamaaBa aaleaacaWG4baabeaaaaa@37E6@  values were set to decrease from age 90 to 110 following an exponential decay.

Rotation of the age patterns of mortality decline

Most models of projecting mortality rates, including the Li-Lee model, keep the age pattern of changes constant over time ( B x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOqamaaBa aaleaacaWG4baabeaaaaa@37E6@ ). However, as Li et al. (2013) note, this has important implications for long-term projections, as the projected age schedule of mortality tends to depart in the long-term from what it should be in light of evolution theory. For example, it is theorized that evolutionary forces shape the age schedule of mortality such that the lowest risk of mortality occurs at the peak reproductive ages. Based on these principles, Li et al. suggested that the curve of (log) mortality rates should keep its checkmark shape, as described earlier in this chapter.

Li et al. note that in the past, rates of mortality decline have been changing, usually performing a "rotation� where rates of changes at older ages accelerated and those at younger ages slowed down. However, these changes are difficult to project. Consisting of second-order differences of the mortality rates, rates of age-specific mortality decline carry larger random fluctuations than changes in the mortality rates themselves. Moreover, Li et al. note that there is not a strong empirical basis that would allow for a data-driven method of accomplishing this task. For these reasons, they suggest to have the B x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOqamaaBa aaleaacaWG4baabeaaaaa@37E6@  structure evolve over time to reach a smoother shape that will help preserve the checkmark shape of mortality rates by age. As a result, the B x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOqamaaBa aaleaacaWG4baabeaaaaa@37E6@  structure becomes flatter and contains less and less information in terms of age heterogeneity over time, as uncertainty grows. The rotational model focuses primarily on the historical declining trend of the ratio or m 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaaBa aaleaacaaIWaaabeaaaaa@37CF@  to m 1519 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaaBa aaleaacaaIXaGaaGynaiabgkHiTiaaigdacaaI5aaabeaaaaa@3AFA@ , but also makes assumptions at other ages (see Li et al. 2013 for more details).

A consequence of the rotational model is that by modifying the pattern of the B x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOqamaaBa aaleaacaWG4baabeaaaaa@37E6@ , the projected life expectancies may change in a somewhat arbitrary way. To prevent this, the K t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBa aaleaacaWG0baabeaaaaa@37EB@  parameters are iterated so that the projected life expectancy remains unchanged in comparison to the results obtained without rotation.Note 15 Hence, the method aims to achieve a more realistic age structure of the projected death rates without modifying life expectancy at birth. Figure 4.11 shows the evolution of the structure of the B x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOqamaaBa aaleaacaWG4baabeaaaaa@37E6@  during the projection.

Figure 4.11 Projected Bx for selected years, by sex, Canada

Description for figure 4.11

Dealing with uncertainty

The model described up to this point has been used for the production of a medium mortality assumption. A low and a high mortality assumption were also built in order to reflect the uncertainty associated with the projection of future mortality. To obtain a plausible confidence interval around the medium assumption, the values for the 80% confidence interval of life expectancy at birth for 2038 estimated by experts in the Opinion Survey on Future Demographic Trends were used as targets for the low and high mortality assumptions. Specifically, the targets were set so that the variation between the medium assumption and the low and high assumptions was the same as that observed in the survey results between the median values for the 80% confidence intervals and the median for the most probable value.Note 16 The life expectancy targets were reached by modifying the K t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBa aaleaacaWG0baabeaaaaa@37EB@ factors, through an iterative process, so that they depart gradually from the K t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBa aaleaacaWG0baabeaaaaa@37EB@  factors of the medium assumption over time.

It should be noted that using the values from the survey ensures a consistent way to handle uncertainty from one component to the next. Moreover, using the survey values provides a reasonable level of uncertainty in the projection in comparison to what is obtained with the Li-Lee model, in which the uncertainty of the whole model is estimated from the variance associated with the projection of the mortality parameters—the time-varying factors. As D'Amato et al. (2011), Liu and Braun (2010), and Koissi et al. (2006) note, the Li-Lee model is expected to underestimate uncertainty as it excludes other sources of uncertainty such as the sampling errors in the parameters.

Although the uncertainty is instilled identically through the K t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBa aaleaacaWG0baabeaaaaa@37EB@  factor in the same provinces and territories, the resulting variations in life expectancy are not identical. In fact, each region has a distinct age-structure of mortality rates that makes their life expectancy at birth more or less sensitive to changes. This is because life expectancy at birth will react more with changes at young ages than at old ages. Thus, regions where mortality rates are relatively high at young ages have more room for improvements at these ages, and generally show more variations in the low and high assumptions than others.

Assumptions

As described earlier, three distinct mortality assumptions were built, representing low, medium and high mortality situations. Figure 4.12 shows the projected age-specific death rates by sex at the Canada level for each assumption at the beginning and end of the projection. It can be seen that the checkmark shape has been preserved over the course of the projection, thanks in large part to changes in the age structure of the rates of mortality decline (the B x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOqamaaBa aaleaacaWG4baabeaaaaa@37E6@  parameter).

Figure 4.12 Projected age-specific death rates at the beginning and at the end of the projection, for the low, medium and high mortality assumptions, by sex, Canada

Description for figure 4.12

In all assumptions, life expectancy at birth at the national level would increase, but at different speeds (Figure 4.13). The increase in life expectancy at birth is projected in all provinces and territories for all assumptions (Tables 4.1, 4.2 and 4.3). The gap in life expectancy at birth between males and females is projected to continue decreasing in all assumptions.

Figure 4.13 Life expectancy at birth, observed (1982 to 2011) and projected (2011/2012 to 2062/2063) as per low, medium and high mortality assumptions, by sex, Canada

Description for figure 4.13

In the medium mortality assumption, the projected life expectancy for males increases from 79.3 years in 2010 to 87.6 years in 2062/2063, while for females it would increase from 83.6 years in 2010 to 89.2 years in 2062/2063 (Table 4.1).

In the low mortality assumption, life expectancy for males is projected to increase from 79.3 years in 2010 to 89.9 years in 2062/2063 while for females it would increase from 83.6 years in 2010 to 91.9 years in 2062/2063 (Table 4.2).

In the high mortality assumption, male life expectancy is projected to grow from 79.3 years in 2010 to 86.0 years in 2062/2063 while for female it would increase from 83.6 years in 2010 to 87.3 years in 2062/2063 (Table 4.3).

Notably, while mortality assumptions were calculated using extrapolation methods, the results for life expectancy at birth either match or are very close to the estimates provided by respondents to the Opinion Survey on Future Demographic Trends. In 2017/2018, the extrapolated values are 80.7 years for males and 84.5 years for females compared to median responses for the most probable estimate from the survey of 80.6 years for males and 84.5 years for females. In 2037/2038, the extrapolated values are 84.3 years for males and 86.8 years for females compared to survey values of 83.9 years for males and 86.6 years for females.

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